A212128 -id:A212128 - cp's OEIS frontend

A344586 Numbers k for which A003415(k) >= A001065(k), where A003415 gives the arithmetic derivative, and A001065 is the sum of proper divisors.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 36, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 100, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117, 120, 121, 124, 125, 127, 128, 131

Offset: 1

Antti Karttunen, May 24 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001065, A003415.
Cf. A212127, A212128 (subsequences), A344585 (complement).
Positions of nonnegative terms in A211991.
Differs from A212165 for the first time at n=121, where a(121) = 220, while A212165(121) = 223.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA344586(n) = (A003415(n) >= (sigma(n)-n));

A238922 Numbers n such that Sum_{i=1..j} 1/d(i) - Sum_{i=1..k} 1/p(i) is an integer, where p are the prime factors of n, counted with multiplicity, and d its divisors.

Original entry on oeis.org

1, 12, 18, 220, 396, 17296, 24016, 287532, 4661056, 64288512, 334144656, 358585488, 555192576, 568719616, 2172649216, 2451538112, 2645953344, 2955423888, 6704333824, 26996772032, 88734733632, 147861504000, 311063879024, 371226582848, 429391876096

Offset: 1

Paolo P. Lava, Mar 07 2014

A212128 and A230164 are subsets of this sequence.

a(26) > 10^12. - Giovanni Resta, Mar 11 2014

			Divisors of 12 are 1, 2, 3, 4, 6, 12 and 1/1 + 1/2 + 1/3 +1/4 + 1/6 + 1/12 = 7/3. Prime factors of 12 are 2^2, 3 and 1/2 + 1/2 + 1/3 = 4/3. Finally 7/3 - 4/3 = 1 that is an integer.

Cf. A003415, A212128, A224346, A230164.

with(numtheory); P:=proc(q) local a,b,c,k,n;
for n from 1 to q do if not isprime(n) then b:=sigma(n)/n;
a:=ifactors(n)[2]; c:=add(a[k][2]/a[k][1],k=1..nops(a));
if type(b-c,integer) then lprint(n,b-c); fi; fi; od; end: P(10^6);

a(9)-a(10), a(13)-a(17), a(19)-a(25) from Giovanni Resta, Mar 11 2014

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A344586 Numbers k for which A003415(k) >= A001065(k), where A003415 gives the arithmetic derivative, and A001065 is the sum of proper divisors.

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A238922 Numbers n such that Sum_{i=1..j} 1/d(i) - Sum_{i=1..k} 1/p(i) is an integer, where p are the prime factors of n, counted with multiplicity, and d its divisors.

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